• Title, Summary, Keyword: Product codes

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Turbo Product Codes Based on Convolutional Codes

  • Gazi, Orhan;Yilmaz, Ali Ozgur
    • ETRI Journal
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    • v.28 no.4
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    • pp.453-460
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    • 2006
  • In this article, we introduce a new class of product codes based on convolutional codes, called convolutional product codes. The structure of product codes enables parallel decoding, which can significantly increase decoder speed in practice. The use of convolutional codes in a product code setting makes it possible to use the vast knowledge base for convolutional codes as well as their flexibility in fast parallel decoders. Just as in turbo codes, interleaving turns out to be critical for the performance of convolutional product codes. The practical decoding advantages over serially-concatenated convolutional codes are emphasized.

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Nonlinear Product Codes and Their Low Complexity Iterative Decoding

  • Kim, Hae-Sik;Markarian, Garik;Da Rocha, Valdemar C. Jr.
    • ETRI Journal
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    • v.32 no.4
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    • pp.588-595
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    • 2010
  • This paper proposes encoding and decoding for nonlinear product codes and investigates the performance of nonlinear product codes. The proposed nonlinear product codes are constructed as N-dimensional product codes where the constituent codes are nonlinear binary codes derived from the linear codes over higher order alphabets, for example, Preparata or Kerdock codes. The performance and the complexity of the proposed construction are evaluated using the well-known nonlinear Nordstrom-Robinson code, which is presented in the generalized array code format with a low complexity trellis. The proposed construction shows the additional coding gain, reduced error floor, and lower implementation complexity. The (64, 24, 12) nonlinear binary product code has an effective gain of about 2.5 dB and 1 dB gain at a BER of $10^{-6}$ when compared to the (64, 15, 16) linear product code and the (64, 24, 10) linear product code, respectively. The (256, 64, 36) nonlinear binary product code composed of two Nordstrom-Robinson codes has an effective gain of about 0.7 dB at a BER of $10^{-5}$ when compared to the (256, 64, 25) linear product code composed of two (16, 8, 5) quasi-cyclic codes.

A NEW CLASS OF CYCLIC CODES USING ORDERED POWER PRODUCT OF POLYNOMIALS

  • Gaur, Ankita;Sharma, Bhudev
    • Journal of applied mathematics & informatics
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    • v.32 no.3_4
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    • pp.529-537
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    • 2014
  • The paper introduces a new product of polynomials defined over a field. It is a generalization of the ordinary product with inner polynomial getting non-overlapping segments obtained by multiplying with coefficients and variable with expanding powers. It has been called 'Ordered Power Product' (OPP). Considering two rings of polynomials $R_m[x]=F[x]modulox^m-1$ and $R_n[x]=F[x]modulox^n-1$, over a field F, the paper then considers the newly introduced product of the two polynomial rings. Properties and algebraic structure of the product of two rings of polynomials are studied and it is shown to be a ring. Using the new type of product of polynomials, we define a new product of two cyclic codes and devise a method of getting a cyclic code from the 'ordered power product' of two cyclic codes. Conditions for the OPP of the generators polynomials of component codes, giving a cyclic code are examined. It is shown that OPP cyclic code so obtained is more efficient than the one that can be obtained by Kronecker type of product of the same component codes.

A Modified Product Code Over ℤ4 in Steganography with Large Embedding Rate

  • Zhang, Lingyu;Chen, Deyuan
    • KSII Transactions on Internet and Information Systems (TIIS)
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    • v.10 no.7
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    • pp.3353-3370
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    • 2016
  • The way of combination of Product Perfect Codes (PPCs) is based on the theory of short codes constructing long codes. PPCs have larger embedding rate than Hamming codes by expending embedding columns in a coding block, and they have been proven to enhance the performance of the F5 steganographic method. In this paper, the proposed modified product codes called MPCs are introduced as an efficient way to embed more data than PPCs by increasing 2r2-1-r2 embedding columns. Unlike PPC, the generation of the check matrix H in MPC is random, and it is different from PPC. In addition a simple solving way of the linear algebraic equations is applied to figure out the problem of expending embedding columns or compensating cases. Furthermore, the MPCs over ℤ4 have been proposed to further enhance not only the performance but also the computation speed which reaches O(n1+σ). Finally, the proposed ℤ4-MPC intends to maximize the embedding rate with maintaining less distortion , and the performance surpasses the existing improved product perfect codes. The performance of large embedding rate should have the significance in the high-capacity of covert communication.

Finite Soft Decision Data Combining for Decoding of Product Codes With Convolutional Codes as Horizontal Codes (길쌈부호를 수평부호로 가지는 곱부호의 복호를 위한 유한 연판정 데이터 결합)

  • Yang, Pil-Woong;Park, Ho-Sung;Hong, Seok-Beom;Jun, Bo-Hwan;No, Jong-Seon;Shin, Dong-Joon
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.37 no.7A
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    • pp.512-521
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    • 2012
  • In this paper, we propose feasible combining rules for a decoding scheme of product codes to apply finite soft decision. Since the decoding scheme of product codes are based on complex tanh calculation with infinite soft decision, it requires high decoding complexity and is hard to practically implement. Thus, simple methods to construct look-up tables for finite soft decision are derived by analyzing the operations of the scheme. Moreover, we focus on using convolutional codes, which is popular for easy application of finite soft decision, as the horizontal codes of product codes so that the proposed decoding scheme can be properly implemented. Numerical results show that the performance of the product codes with convolutional codes using 4-bit soft decision approaches to that of same codes using infinite soft decision.

Enhanced Upper Bound for Erasure Recovery in SPC Product Codes

  • Muqaibel, Ali
    • ETRI Journal
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    • v.31 no.5
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    • pp.518-524
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    • 2009
  • Single parity check (SPC) product codes are simple yet powerful codes that are used to correct errors and/or recover erasures. The focus of this paper is to evaluate the performance of such codes under erasure scenarios and to develop a closed-form tight upper bound for the post-decoding erasure rate. Closed-form exact expressions are derived for up to seven erasures. Previously published closed-form bounds assumed that all unrecoverable patterns should contain four erasures in a square. Additional non-square patterns are accounted for in the proposed expressions. The derived expressions are verified using exhaustive search. Eight or more erasures are accounted for by using a bound. The developed expressions improve the evaluation of the recoverability of SPC product codes without the need for simulation or search algorithms, whether exhaustive or novel.

On Combining Chase-2 and Sum-Product Algorithms for LDPC Codes

  • Tong, Sheng;Zheng, Huijuan
    • ETRI Journal
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    • v.34 no.4
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    • pp.629-632
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    • 2012
  • This letter investigates the combination of the Chase-2 and sum-product (SP) algorithms for low-density parity-check (LDPC) codes. A simple modification of the tanh rule for check node update is given, which incorporates test error patterns (TEPs) used in the Chase algorithm into SP decoding of LDPC codes. Moreover, a simple yet effective approach is proposed to construct TEPs for dealing with decoding failures with low-weight syndromes. Simulation results show that the proposed algorithm is effective in improving both the waterfall and error floor performance of LDPC codes.

Erasure decoding strategies for RS product code reducing undetected error rate (검출 불능 오류율을 향상기키는 Reed-Solomon 적부호의 이레이져 복호방법)

  • 김정헌;염창열;송홍엽;강구호;김순태;백세현
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.26 no.4B
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    • pp.427-436
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    • 2001
  • RS product codes are widely used in digital storage systems. There are lots of decoding strategies for product code for short-length RS codes. Unfortunately many of them cannot be applied to long-length RS product codes because of the complexity of decoder. This paper proposes new decoding strategies which can be used in long length RS product codes.

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ADDITIVE SELF-DUAL CODES OVER FIELDS OF EVEN ORDER

  • Dougherty, Steven T.;Kim, Jon-Lark;Lee, Nari
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.2
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    • pp.341-357
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    • 2018
  • We examine various dualities over the fields of even orders, giving new dualities for additive codes. We relate the MacWilliams relations and the duals of ${\mathbb{F}}_{2^{2s}}$ codes for these various dualities. We study self-dual codes with respect to these dualities and prove that any subgroup of order $2^s$ of the additive group is a self-dual code with respect to some duality.

Performance Analysis of Turbo Product Code Using Parallel Structure (병렬 구조를 이용한 Turbo Product Code 성능 분석)

  • 이태길;정지원
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.29 no.2C
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    • pp.181-186
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    • 2004
  • Recently, there has been intensive focus on Turbo Product Codes(TPCs) which have low decoding complexity and achieve near-optimum performances at high code-rate. This paper present a parallel algorithm of turbo product codes enable simultaneous decoding of row and column. The row and column decoders operate in parallel and update each other after row and column has been decoded. simulation results show that the performance of proposed parallel turbo code is almost the same as that conventional scheme for several turbo product codes.